Maximum Entropy, Information Without Probability and Complex Fractals

Jumarie, Guy

doi:10.1007/978-94-015-9496-7

Cited by 47 publications

(56 citation statements)

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“…This ensemble differs from the GOE and from the other known ensembles of the random matrix theory. This new ensemble is constructed using the maximum entropy principle [31][32][33][34] which allows the probability distribution of positive symmetric real random matrices to be constructed using only the available information. In order to improve the readability of this paper, one recalls fundamentals of the non-parametric model of random uncertainties introduced in the papers mentioned above.…”

Section: Introductionmentioning

confidence: 99%

Random matrix theory and non-parametric model of random uncertainties in vibration analysis

Soize¹

2003

Journal of Sound and Vibration

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To cite this version:Christian Soize. Random matrix theory and non-parametric model of random uncertainties in vibration analysis. Journal of Sound and Vibration, Elsevier, 2003, 263 (4) of random matrices constituted of symmetric positive-definite real random matrices. This ensemble differs from the GOE and from the other known ensembles of the random matrix theory. The present paper has three main objectives. The first one is to study the statistics of the random eigenvalues of random matrices belonging to this new ensemble and to compare with the GOE. The second one is to compare this new ensemble of random matrices with the GOE in the context of the non-parametric approach of random uncertainties in structural dynamics for the low-frequency range. The last objective is to give a new validation for the non-parametric model of random uncertainties in structural dynamics in comparing, in the low-frequency range, the dynamical response of a simple system having random uncertainties modelled by the parametric and the non-parametric methods. These three objectives will allow us to conclude about the validity of the different theories.

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Section: Introductionmentioning

confidence: 99%

Random matrix theory and non-parametric model of random uncertainties in vibration analysis

Soize¹

2003

Journal of Sound and Vibration

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“…It suffices to start out with a general divergence function on X ⊗ Y in order for the construction to make sense. When the construction is based on a general divergence function D, we refer to (14) as the updating triple generated by D and with y 0 as prior. For these updating triples, we take y 0 as the only certain belief instance.…”

Section: Relativization Updatingmentioning

confidence: 99%

“…To emphasize this, we introduce, based only on a general divergence function D, the effort-based information triple associated with (14) as the triple…”

Section: Relativization Updatingmentioning

confidence: 99%

“…Many areas within psychology are also of relevance. Some specific works of interest include Jumarie [14], Shafer and Vovk [15], Gernert [16], Bundesen and Habekost [17], Benedetti [18] and Brier [19]. The handbook [20] edited by Adriaans and Bentham and the encyklopedia article [21] by Adriaans collect views on the very concept of information.…”

Section: Introductionmentioning

confidence: 99%

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A Quantitative Theory of Cognition with Applications

Topsøe

2014

Proceedings of 1st International Electronic Conference on Entropy and Its Applications

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Cognition is based partly on belief and strives for knowledge of "truth". Description and the derived notion of control lead to notions of entropy and divergence and help to clarify what can be known, the "knowable", by setting limits to information. Via game theoretical considerations, a notion of core is introduced which, in classical settings, is related to that of exponential families. It facilitates the process of inference in concrete cases of interest. The basic setting is abstract. When we turn to probabilistic modelling, interaction between truth, belief and knowledge is essential for an interpretation of Tsallis entropy.

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“…In Refs. [74,75,81], we introduced the ensemble SG + of the random matrices [G n ], defined on a probability space (Θ, T , P ) with values in Å + n (Ê), whose probability distribution was constructed by using the entropy optimization principle [65,34,36,37] C. Soize -CMAME -revised version December 2004 with the constraints defined by the following available information: [G n ] is a symmetric positivedefinite real random matrix whose mean value is the identity matrix and for which the second-order moment of the random variable [G n ] −1 F is finite. The probability distribution P [G n ] on Å + n (Ê) of such a random matrix [G n ] was explicitly constructed and depends only on dimension n and a positive real parameter δ independent of n and allowing the dispersion of random matrix [G n ] to be controlled.…”

Section: Principle Of the Constructionmentioning

confidence: 99%

Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators

Soize¹

2006

Computer Methods in Applied Mechanics and Engineering

162

235

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To cite this version:Christian Soize. Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2006, 195 (1-3) AbstractThis paper deals with the construction of a class of non Gaussian positive-definite matrix-valued random fields whose mathematical properties allow elliptic stochastic partial differential operators to be modeled. The properties of this class is studied in details and the numerical procedure for constructing numerical realizations of the trajectories is explicitly given. Such a matrix-valued random field can directly be used for modeling random uncertainties in computational sciences with a stochastic model having a small number of parameters. The class of random fields which can be approximated is presented and their experimental identification is analyzed. An example is given in three-dimensional linear elasticity for which the fourth-order elasticity tensor-valued random field is constructed for a random non homogeneous anisotropic elastic material.

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Maximum Entropy, Information Without Probability and Complex Fractals

Cited by 47 publications

References 0 publications

Random matrix theory and non-parametric model of random uncertainties in vibration analysis

Random matrix theory and non-parametric model of random uncertainties in vibration analysis

A Quantitative Theory of Cognition with Applications

Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators

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